Modeling complex systems by simple mathematical models using by yaofenji

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									Representing complex engineering systems by
simple mathematical models



                         May 23, 2007
                         Jason Mayes
Outline

    Motivation
        Mathematical modeling
        Complex systems
    Objectives
    Present Work
        Using self-similarity
        Using fractional-order system identification
    Future Work
Mathematical Modeling
   A mathematical model is a mathematical description of a
    physical system or process
        Design
        Prediction
        Control
   Mathematical modeling: finding a compromise between
    an intractable problem and a model that sufficiently
    describes the system
   Characteristics of a desirable model
        Simple
        Adequately describes the system
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Mathematical Modeling
                                                                    x(t)
         • Electro-mechanical model
         • Equations of motion
         • Add complexity                                      Fg
         • Experiments


                            Random Experimental Data

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The philosophy of analysis
   Many ways to find a mathematical model
Methodological Reductionism
   Descartes‟ 1637 Discourse on Method
        The world is like a machine…
        Everything can be reduced to many smaller, simpler things
   The best way to understand a system is to first gain a clear
    understanding of its smallest subsystems
   Models are based on first principles
   Problems:
        Size and complexity
Holism
   Behavior must be studied on the level of the system as a whole

   Aristotle‟s Metaphysics –
                 “the whole is more than the sum of its parts”
                                                           Random Experimental Data



   Examples                                 1.2


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      Neural nets
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      On/off, PID, fuzzy logic
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      Expert systems
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   Black-box analysis is useful for describing or controlling
    systems, but it doesn‟t explain observed behavior
A model-based compromise
   Mix of holistic and reductionist approaches
   Model-based parameter identification
   Can form a model containing a few free parameters
   Very common in heat transfer                N
        Nusselt number correlations, convection        F  N
        Contact resistance, fouling coefficients
        Heat exchangers
        Friction factor (Moody diagram)                  F



   Trend in science: HolisticReductionist analysis
        Example: chemical reactions
Complex systems
   Q: What is a complex system?
    A: ???
       Properties:
            Size/complexity
            Non-linear
            Emergent phenomena
            Memory

   Our working definition: complex engineering systems
       Physically: any system composed of a large number of components
        and interactions that creates difficulties in both understanding and
        modeling
       Mathematically: a large system of coupled equations which are
        either too complex or too large to admit a sufficiently useful
        solution
Objectives
    To find simple mathematical models of complex
     engineering systems
        Using both reductionist and holistic approaches to modeling

    Systems with similarity
        Both physical and mathematical similarity
        Can reduce complex mathematical models to simpler models

    Using fractional-order system identification
        Modeling complex systems as a black-box
        Better results than traditional integer-order models
Mathematical self-similarity
   Current Work: using a reductionist approach and taking advantage of
    mathematical similarity to simplify complex models


   Mathematical similarity:
        Equations of the same form
        Repeating patterns or coupling in equations


   Reducible mathematical models
        Complex system: large system of ODEs
        System of first order ODEs  scalar ODE
        High-order scalar ODE  lower order scalar ODE
Mathematical Similarity
        Reducing infinite-order ODEs

   High-order ODEs can be reduced in the Laplace domain




               Geometric Series!!
    Mathematical Similarity
           Reducing infinite-order ODEs

   Can reduce an infinite-order (very large) ODE to a simple,
    finite-order ODE




   Only useful in complex mechanical systems if infinite-order
    ODEs occur in modeling
Mathematical Similarity
          Reducing infinite sets of ODEs

   High-order (infinite) ODEs result
    from large (infinite) equations sets
Mathematical Similarity
        Reducing infinite sets of ODEs

   Consider a very large system of springs and masses
    Mathematical Similarity
            Reducing PDEs


   Infinite sets of ODEs also result from the reduction of PDEs
       Finite-volume
       Spectral methods
       Finite difference


   Example: heat equation
      Mathematical Similarity
             Reducing PDEs


   Finite-volume formulation
    Mathematical Similarity
          Reducing PDEs

   An infinite set of differential equations
Mathematical Similarity
        Reducing PDEs

   Can now reduce the continued fraction




   Taking the limit

   Now take the inverse Laplace transform
    Mathematical Similarity
              Reducing PDEs

       Reduced the PDE in a way that we have extracted the heat
        flux at the boundary



       Reduction process:

PDE  System of ODEs  Single high-order ODE  Single low-order ODE

       Alternative:
            Solve PDE
            Get a „global‟ solution
            Differentiate at the boundary
Mathematical Similarity
          Applications of PDE reduction

   Only need a „local‟ solution
   Change of boundary conditions
   Blast furnace monitoring [Oldham and Spanier, 1974]
   Laser or cryogenic surgery
        Heat equation [Taler, 1996]
        Need only one thermocouple
    Physical self-similarity

   Current work: using physical self-
    similarity to reduce complex models

   Potential driven flows through
    bifurcating trees

   Self-similar equations sets can also
    result from physically self-similar
    systems
          Physical Similarity
                   A self-similar model


   The bifurcating tree geometry
        Geometry seen in a wide variety of
         applications [Bejan, 2000]
   Potential-driven flow or transfer
        ex. heat, fluid, energy, etc.
   Conservation at bifurcation points

                                  q1

         q

                q = q1+q2         q2
Physical Similarity
         A self-similar model

   Assumptions
       Conservation at nodes
       Transfer governed by a linear operator
       i.e., for each branch:


       Large system of DAE‟s
            2n+1 -2 differential „branch‟ equations
            2n-1 continuity equations
Physical Similarity
         Reduction

   System of DAEs can be reduced as before
   Regular coupling from physical similarity allows for reduction
   For N=1 generation network:




                 }
Physical Similarity
       Reduction

   For N-generation network
         Physical Similarity
                     Additional forms of similarity

   Similarity in operators can be used to further simplify
        Two forms of similarity:
             Similarity „within‟ a generation
                   Symmetric networks: the operators within a generation are identical
                   Asymmetric networks: the operators within a generation are not identical
             Similarity „between‟ generations
                   Generation dependent operators depend on the generation in which the operator
                    occurs and change between successive generations
                   Generation independent operators do not change between generations
   Four possible combinations
        Symmetric with generation independent operators
        Asymmetric with generation independent operators
        Symmetric with generation dependent operators
        Asymmetric with generation dependent operators
     Physical Similarity
              Symmetric and generation independent

         Can further reduce the continued fraction




For infinite networks:
     Physical Similarity
              Asymmetric and generation independent

         Further reduction:




For infinite networks:
      Physical Similarity
                  Symmetric and generation dependent

          Further reduction:




For infinite networks:
• Can further reduce for
certain forms of
      Physical Similarity
                Asymmetric and generation dependent

         Further reduction:
              No general reduction
              Can reduce further for specific cases




For infinite networks:
         Physical Similarity
                 Applications

   Viscoelasticity
         Asymmetric, generation independent
         Fractional-order viscoelastic models
         Springs (k) and dampers (µ)
         [Heymans and Bauwens, 1994]
         Physical Similarity
                 Applications

   Laminar flow through bifurcating trees
         Symmetric, generation dependent
         Using laminar pipe flow model for
          each branch:
Using Similarity
         Summary of current work

   Representing complex engineering systems by
    simple mathematical models

       From a reductionist perspective:
            Mathematical systems with similarity
                  PDEs  Systems of ODEs  High-order scaler ODE  Lower-order scaler ODE

            Physical systems with similarity
                  Use physical structure to reduce model
                  Additional forms of similarity can offer further reduction
Fractional-order system ID
   Current work: using fractional-order models to better fit nearly
    exponential experimental data
   System identification: building a dynamic mathematical model of a
    system using measured/observed data
   Holistic approach to modeling
   Grey- or black-box modeling
   Typically assumes integer-order models
   Fractional-order models can
    often do a better job describing
    real physical systems
    [Podlubny, 1999]


                                               ?
Fractional-order system ID
         Nearly exponential transitions

   Focus on nearly exponential transitions
   A transition from one steady-state to
    another, usually the result of step change
    in input
   Typically assumed to be some combination
    of exponentials
   Commonly modeled as first or second
    order systems:




   Fractional-order models can often do a better job
Fractional-order system ID
            Linear systems: a toy problem

     Consider the system




     System representation:

    U (s)             6           Y (s)
               s  6s  11s  6
                3    2




     System identification:
Fractional-order system ID
          Linear systems: a fractance device

   Fractance device – an electrical circuit having properties which lie
    between resistance and capacitance or resistance and inductance.
                                  
   Has an impedance Z  (i )

   Construction: infinite bifurcating network of resistors and capacitors
    [Nakagawa and Sorimachi, 1994]


        N = 1:




        N = ∞:
Fractional-order system ID
         Linear systems: a fractance device

   For N=1, 3, 6, and 9 generation fractance devices



           9
         N=1
                                         Fractional-order model is a special
                                         case of the integer-order model
Fractional-order system ID
            Nonlinear systems: a toy problem

     Consider the system




     System representation:


    U (s)                      Y (s)


     System identification:
Fractional-order system ID
          Nonlinear systems: experimental results

   Shell-and-tube heat exchanger
   Steady state correlations are
    readily available
   Interested in a dynamic correlation
    for the response to a step input
   Complex systems: reductionist approach
    leads a complex model
        System of PDEs
        Turbulence, recirculation
        Very detailed
   Holistic or black-box: experimentally determine response
        Measured hot-side outlet temperature

        Step change in cold-side inlet flow rate
Fractional-order system ID
          Nonlinear systems: experimental results

   Experimental results:
Fractional-order system ID
         Summary of current work

   Representing complex engineering systems by simple
    mathematical models
       From a holistic perspective:
       Fractional-order models often better descriptors of complex
        systems than traditional integer-order models
            Integer-order models are special cases of fractional-order
             models
            Useful for modeling linear and non-linear systems
       Heat exchanger shown to be modeled well using a
        fractional-order model [Aoki, 2006]
Future Work
   Work to proceed in two directions
       Reductionism
            Finding simple mathematical models of complex engineering
             systems that exhibit similarity properties
       Holism
            Using simple fractional-order models
Future Work
        Reduction using similarity


   Reducing PDEs
       Approximate boundary condition
        transformations
            Pennes equation



       2-D PDEs
Future Work
        Reduction using similarity


   Reducing physical systems
       Grid-like transfer networks
Future Work
         Reduction using fractional-order models

   Using fractional-order models
        Continue heat exchanger experiments to better develop
         dynamic correlation
        Fractional-order control:

             Fractional-order systems can be controlled more efficiently using
              fractional-order control techniques
             Using fractional-order system identification and experimental
              results to choose proper controller gains
                   Similar to Ziegler-Nichols method of parameter tuning
                   Based on offline time response characteristics
Questions?

								
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